A045572 -id:A045572 - cp's OEIS frontend

A045808 8-ish numbers (end in 67, 69, 81, 83).

67, 69, 81, 83, 167, 169, 181, 183, 267, 269, 281, 283, 367, 369, 381, 383, 467, 469, 481, 483, 567, 569, 581, 583, 667, 669, 681, 683, 767, 769, 781, 783, 867, 869, 881, 883, 967, 969, 981, 983, 1067, 1069, 1081, 1083, 1167, 1169, 1181, 1183, 1267, 1269

Offset: 1

J. H. Conway

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A045800-A045809.

Cf. A045572, A045797, A045798.

import Data.List (findIndices)
a045808 n = a045808_list !! (n-1)
a045808_list = findIndices (`elem` [67,69,81,83]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

Select[Range[1300],MemberQ[{67,69,81,83},Mod[#,100]]&] (* Harvey P. Dale, Jul 03 2013 *)

Empirical g.f.: x*(67+2*x+12*x^2+2*x^3+17*x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012

More terms from Erich Friedman

A175545 Numbers n (relatively prime to 10) such that the decimal form of the period of 1/n is prime.

Original entry on oeis.org

3, 27, 33, 333, 369, 909, 2151, 2439, 2997, 3333, 27027, 33333, 37683, 41841, 76923, 90909, 142857, 194841, 243603, 333333

Offset: 1

Michel Lagneau, Jun 24 2010

This sequence is infinite because the numbers 3, 33, 333, ... generate the decimal form 3. The correspondant primes of this sequence such that :
{3, 37, 3, 3, 271, 11, 4649, 41, 333667, 3} are included in the sequence A178505.
The Maple program below is very slow for the numbers > 3333.

			27 is in the sequence because 1/27 = 0.037 037 ... and 37 is prime.
2997 is in the sequence because 1/2997 = 0.000333667 000333667 ... and 333667 is prime.

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrueche'.

Index entries for sequences related to decimal expansion of 1/n.

Cf. A178505, A045572, A002329.

with(numtheory): Digits:=4000:nn:=4000:for n from 3 by 2 to nn do:z:=evalf(1/n): indic:=0:for p from 1 to nn do:if irem(10^p, n) = 1 and gcd(n, 5) = 1 and indic=0 then pp:=p:indic:=1:z1:=floor(z*10^pp): else fi:od:if indic=1 and type(z1,prime)=true then print(n):else fi:od:

Extended and name corrected by T. D. Noe, Nov 18 2010

a(17)-a(20) from Ray Chandler, Apr 17 2017

A175550 Period of the decimal expansion of 1/F as F runs through the Fibonacci numbers greater than 1 and not divisible by 2 or 5.

Original entry on oeis.org

1, 6, 6, 44, 232, 84, 138, 133, 336, 396, 28656, 3016, 84, 514228, 335824, 152214, 67830, 4440, 261744, 504628, 108373609, 47124, 3295440, 2971215072, 49349664, 45240, 4438362040, 203028, 3599596, 10841042784, 104340657248, 252736776688

Offset: 1

Michel Lagneau, Jun 26 2010

The Fibonacci numbers contributing to this sequence are {3, 13, 21, 89, 233, ...}, i.e., Fibonacci(k) for k = 4, 7, 8, 11, 13, ... (A229829, starting with A229829(3)).

			For n = 1, the 1st Fibonacci number > 1 and coprime to 2 and 5 is Fibonacci(4) = 3, and period(1/3) = 1, so a(1) = 1.
For n = 2, the 2nd Fibonacci number > 1 and coprime to 2 and 5 is Fibonacci(7) = 13, and period (1/13) = 6, so a(2) = 6.

Cf. A000045, A175561, A178505, A045572, A002329, A229829.

with(combinat, fibonacci):nn:= 50:for q from 1 to nn do:n:=fibonacci(q):indic:=0:for p from 1 to n do:if irem(10^p, n) = 1 and gcd(n, 5) = 1 and indic=0 then printf(`%d, `, p):indic:=1:else fi:od:od:

Table[MultiplicativeOrder[10, n/Times @@ ({2, 5}^IntegerExponent[n, {2, 5}])], {n, Select[Fibonacci[Range[3, 70]], CoprimeQ[#, 10] &]}] (* Amiram Eldar, May 27 2024 *)

a(15) onwards from Robert G. Wilson v, Jun 29 2010

A267755 Expansion of (1 + 2x + x^2 + x^3 + 4x^4 + 2*x^5)/(1 - x - x^5 + x^6).

Original entry on oeis.org

1, 3, 4, 5, 9, 12, 14, 15, 16, 20, 23, 25, 26, 27, 31, 34, 36, 37, 38, 42, 45, 47, 48, 49, 53, 56, 58, 59, 60, 64, 67, 69, 70, 71, 75, 78, 80, 81, 82, 86, 89, 91, 92, 93, 97, 100, 102, 103, 104, 108, 111, 113, 114, 115, 119, 122, 124, 125, 126, 130, 133, 135, 136, 137

Offset: 0

Bruno Berselli, Jan 20 2016

(m^k-1)/11 is a nonnegative integer when
. m is a member of this sequence and k is an odd multiple of 5 (A017329),
. m is a member of A017401 and k is odd but not multiple of 5 (A045572),
. m is a member of A175885 and k is even but not multiple of 5 (A217562),
. m is a member of A160542 and k is a positive multiple of 10 (A008592),
apart from the trivial case in which k=0.
Also, numbers that are congruent to {1, 3, 4, 5, 9} mod 11. Therefore, the product of two terms belongs to the sequence.
Union of this sequence and A267541 is A160542.
a(n) is prime for n = 1, 3, 10, 14, 17, 21, 24, 27, 30, 33, 40, 44, 47, ...

			From the linear recurrence:
(-A267541) ..., -13, -10, -8, -7, -6, -2, 1, 3, 4, 5, 9, 12, ... (A267755)

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A160542, A017329, A267541.

Related sequences (see the first comment): A017401, A160542, A175885.

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+x^2+x^3+4*x^4+2*x^5)/(1-x-x^5+x^6)));

I:=[1,3,4,5,9,12]; [n le 6 select I[n] else Self(n-1)+Self(n-5)-Self(n-6): n in [1..70]]; // Vincenzo Librandi, Jan 21 2016

gf := (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6): deg := 64: series(gf, x, deg): seq(coeff(%, x, n), n=0..deg-1); # Peter Luschny, Jan 21 2016

CoefficientList[Series[(1 + 2 x + x^2 + x^3 + 4 x^4 + 2 x^5)/(1 - x - x^5 + x^6), {x, 0, 70}], x]
LinearRecurrence[{1, 0, 0, 0, 1, -1}, {1, 3, 4, 5, 9, 12}, 70]
Select[Range[140], MemberQ[{1, 3, 4, 5, 9}, Mod[#, 11]]&]

Vec((1+2*x+x^2+x^3+4*x^4+2*x^5)/(1-x-x^5+x^6)+O(x^70))

gf = (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6)
print(taylor(gf, x, 0, 63).list()) # Peter Luschny, Jan 21 2016

G.f.: (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6).
a(-n) = -A267541(n-1).
a(n) = n + 1 + 2*floor(n/5) + 3*floor((n+1)/5) + floor((n+4)/5). - Ridouane Oudra, Sep 06 2023

A348480 For numbers x_n coprime to 10 there exist infinitely many binary numbers b such that gcd(b,rev(b)) = x_n and digitsum(b) = x_n. a(n) is the smallest b converted to decimal that satisfies this constraint.

Original entry on oeis.org

1, 11, 4399137296449, 767, 4543829, 302306413101798081695809, 1041919, 4120511, 119471087, 92239871, 461373439, 3221191679, 25098711039, 5864072675327, 2642508222647189060948556167549513, 20016007615544303, 208836273045503, 70085007900671, 985162418485119

Offset: 1

Ruediger Jehn, Oct 20 2021

Only for numbers x_n coprime to 10 (A045572, i.e., numbers ending with 1,3,7 or 9) do there exist binary numbers b such that gcd(b, rev(b)) = x_n and digitsum(b) = x_n. For the numbers 7 and 13 and the porous numbers 11, 37 and 101 (A337832), the terms in their binary form have more zeros than ones, which are called long solutions. In these cases, let e = mult_order(10, n), then b = 10^(e*n) + Sum_{i=0..n-2} 10^(e*i). For example, the multiplicative order of 10 mod 11 is 2 and 10001010101010101010101 is the solution. However, in the case of the porous number 121, this formula does not work because both b and rev(b) are divisible by 1111111111111111111111 which also has a multiplicative order of 10 = 22 like 121 and therefore two extra zeros need to be inserted.
For most numbers short solutions exist. Which numbers have a short solution and which have a long solution is still unclear.
For clarification: in gcd(1011,1101)=3 the two numbers 1011 and 1101 are base-10 numbers, but then 1011 is interpreted as a base-2 number and translated back to base 10 to get a(2)=11 (=8+2+1).

			x_2 = 3. a(2)=11 which in binary is 1011. gcd(1011,1101)=3 and there is no smaller binary number that satisfies this constraint.
x_4 = 9. a(4)=767 which in binary is 1011111111. gcd(1011111111,1111111101)=9 and there is no smaller binary number that satisfies this constraint.

Ruediger Jehn, Table of n, a(n) for n = 1..54
Rüdiger Jehn, A new 200 Euro math puzzle, Youtube video, Sep 17 2021.
Rüdiger Jehn, Long Solutions of Sequence A348480 of the On-Line Encyclopedia of Integer Sequences, arXiv:2201.00710 [math.GM], 2022.

Cf. A045572, A333666, A337832, A350385.

xx(n) = 2*n - 1 + (n+1)\4 * 2; \\ A045572
gcdr(n) = my(b=binary(n)); gcd(fromdigits(Vecrev(b), 10), fromdigits(b, 10));
a(n) = my(b=1, x=xx(n)); while ((hammingweight(b) != x) || (gcdr(b) != x), b++); b; \\ Michel Marcus, Dec 01 2021

from sympy.utilities.iterables import multiset_permutations
from itertools import count
from math import gcd
def A348480(n):
    if n == 1: return 1
    xn = 2*(n+(n+1)//4) - 1
    for l in count(xn-1):
        for d in multiset_permutations(['0']*(l-xn+1)+['1']*(xn-1)):
            s = '1'+''.join(d)
            if gcd(int(s),int(s[::-1])) == xn:
                return int(s,2) # Chai Wah Wu, Jan 08 2022

a(13) from Giorgos Kalogeropoulos, Oct 22 2021
a(14) from Pontus von Brömssen, Oct 23 2021
a(15) from Ruediger Jehn, Dec 01 2021
a(16) - a(29) from Ruediger Jehn, Dec 17 2021
a(30) - a(54) from Ruediger Jehn, Jan 11 2022

A045802 2-ish numbers (end in 03, 21, 29, 47).

Original entry on oeis.org

3, 21, 29, 47, 103, 121, 129, 147, 203, 221, 229, 247, 303, 321, 329, 347, 403, 421, 429, 447, 503, 521, 529, 547, 603, 621, 629, 647, 703, 721, 729, 747, 803, 821, 829, 847, 903, 921, 929, 947, 1003, 1021, 1029, 1047, 1103, 1121, 1129, 1147, 1203, 1221

Offset: 1

J. H. Conway

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A045800-A045809, A045572, A045797, A045798.

import Data.List (findIndices)
a045802 n = a045802_list !! (n-1)
a045802_list = findIndices (`elem` [3,21,29,47]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

Join[{3},Select[Range[10,1300],MemberQ[{{0,3},{2,1},{2,9},{4,7}}, Take[ IntegerDigits[#],-2]]&]] (* or *) LinearRecurrence[{1,0,0,1,-1},{3,21,29,47,103},50] (* Harvey P. Dale, May 05 2012 *)

G.f.: x*(3+18*x+8*x^2+18*x^3+53*x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012
a(n) = (50*n-24*i^(n*(n+1))-7*(-1)^n-75)/2, where i=sqrt(-1). - Bruno Berselli, Feb 23 2012
a(1)=3, a(2)=21, a(3)=29, a(4)=47, a(5)=103, a(n)=a(n-1)+a(n-4)-a(n-5) .[Harvey P. Dale, May 05 2012]

More terms from Erich Friedman.

A045803 3-ish numbers (end in 17, 19, 31, 33).

Original entry on oeis.org

17, 19, 31, 33, 117, 119, 131, 133, 217, 219, 231, 233, 317, 319, 331, 333, 417, 419, 431, 433, 517, 519, 531, 533, 617, 619, 631, 633, 717, 719, 731, 733, 817, 819, 831, 833, 917, 919, 931, 933, 1017, 1019, 1031, 1033, 1117, 1119, 1131, 1133, 1217, 1219

Offset: 1

J. H. Conway

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A045800-A045809.

Cf. A045572, A045797, A045798.

import Data.List (findIndices)
a045803 n = a045803_list !! (n-1)
a045803_list = findIndices (`elem` [17,19,31,33]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

Select[Range[1300],MemberQ[{17,19,31,33},Mod[#,100]]&] (* or *) LinearRecurrence[{1,0,0,1,-1},{17,19,31,33,117},50] (* Harvey P. Dale, Dec 17 2014 *)

a(n) = -75/2 - (23*(-1)^n)/2 - (9-9*I)*(-I)^n - (9+9*I)*I^n + 25*n \\ Colin Barker, Oct 16 2015

Vec(x*(17+2*x+12*x^2+2*x^3+67*x^4)/(1-x-x^4+x^5) + O(x^100)) \\ Colin Barker, Oct 16 2015

G.f.: x*(17+2*x+12*x^2+2*x^3+67*x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012

a(n) = -75/2 - (23*(-1)^n)/2 - (9-9*i)*(-i)^n - (9+9*i)*i^n + 25*n where i=sqrt(-1). - Colin Barker, Oct 16 2015

More terms from Erich Friedman

A045804 4-ish numbers (end in 09, 41, 63, 87).

Original entry on oeis.org

9, 41, 63, 87, 109, 141, 163, 187, 209, 241, 263, 287, 309, 341, 363, 387, 409, 441, 463, 487, 509, 541, 563, 587, 609, 641, 663, 687, 709, 741, 763, 787, 809, 841, 863, 887, 909, 941, 963, 987, 1009, 1041, 1063, 1087, 1109, 1141, 1163, 1187, 1209, 1241

Offset: 1

J. H. Conway

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A045800-A045809.

Cf. A045572, A045797, A045798.

import Data.List (findIndices)
a045804 n = a045804_list !! (n-1)
a045804_list = findIndices (`elem` [9,41,63,87]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

Select[Range[1250],MemberQ[{9,41,63,87},Mod[#,100]]&] (* Harvey P. Dale, Aug 01 2020 *)

G.f.: x*(9+32*x+22*x^2+24*x^3+13*x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012

More terms from Erich Friedman.

A045805 5-ish numbers (end in 51, 57, 93, 99).

Original entry on oeis.org

51, 57, 93, 99, 151, 157, 193, 199, 251, 257, 293, 299, 351, 357, 393, 399, 451, 457, 493, 499, 551, 557, 593, 599, 651, 657, 693, 699, 751, 757, 793, 799, 851, 857, 893, 899, 951, 957, 993, 999, 1051, 1057, 1093, 1099, 1151, 1157, 1193, 1199, 1251, 1257

Offset: 1

J. H. Conway

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A045800-A045809.

Cf. A045572, A045797, A045798.

import Data.List (findIndices)
a045805 n = a045805_list !! (n-1)
a045805_list = findIndices (`elem` [51,57,93,99]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

Select[Range[50,1500],MemberQ[{51,57,93,99},Mod[#,100]]&] (* Harvey P. Dale, Nov 14 2022 *)

Empirical g.f.: x*(51+6*x+36*x^2+6*x^3+x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012

More terms from Erich Friedman

A045806 6-ish numbers (end in 23, 27, 61, 89).

Original entry on oeis.org

23, 27, 61, 89, 123, 127, 161, 189, 223, 227, 261, 289, 323, 327, 361, 389, 423, 427, 461, 489, 523, 527, 561, 589, 623, 627, 661, 689, 723, 727, 761, 789, 823, 827, 861, 889, 923, 927, 961, 989, 1023, 1027, 1061, 1089, 1123, 1127, 1161, 1189, 1223, 1227

Offset: 1

J. H. Conway

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A045800-A045809.

Cf. A045572, A045797, A045798.

import Data.List (findIndices)
a045806 n = a045806_list !! (n-1)
a045806_list = findIndices (`elem` [23,27,61,89]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

Empirical g.f.: x*(23+4*x+34*x^2+28*x^3+11*x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012

More terms from Erich Friedman

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A045808 8-ish numbers (end in 67, 69, 81, 83).

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A175545 Numbers n (relatively prime to 10) such that the decimal form of the period of 1/n is prime.

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A175550 Period of the decimal expansion of 1/F as F runs through the Fibonacci numbers greater than 1 and not divisible by 2 or 5.

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A267755 Expansion of (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6).

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A348480 For numbers x_n coprime to 10 there exist infinitely many binary numbers b such that gcd(b,rev(b)) = x_n and digitsum(b) = x_n. a(n) is the smallest b converted to decimal that satisfies this constraint.

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A045802 2-ish numbers (end in 03, 21, 29, 47).

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A045803 3-ish numbers (end in 17, 19, 31, 33).

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A267755 Expansion of (1 + 2x + x^2 + x^3 + 4x^4 + 2*x^5)/(1 - x - x^5 + x^6).